Conditionals

1
Conditionals

• Section 1.2

2
Conditionals

• One statement logically implies another if from
  the truth of the one we can infer the truth of
  the other by virtue solely of the logical
  structure of the two statements
• In other words, when you make a logical
  deduction, you reason from a hypothesis to a
  conclusion
• If you show up for work Monday morning, then you
  will get the job
• If p then q
• If (hypothesis) then (conclusion)
• The truth statement of q is conditioned on the
  truth of statement p

3
Implication p ? q

• Let p and q be propositions. The proposition p
  implies q is denoted p ? q
• False when p is true and q is false
• True otherwise
• p is called the hypothesis and q is called the
  conclusion

4
Implication Example

• If Roland eats sweets then he will get fat
• p Roland eats sweets
• q Roland gets fat
• p true and q true
• Roland eats sweets and gets fat (True)
• p true and q false
• Roland eats sweets and does not get fat (False)
• p false and q true
• Roland does not eat sweets but does get fat
  (True)
• Proposition does not say what will happen if
  Roland did not eat sweets
• p false and q false
• Roland does not eat sweets and does not get fat
  (True)
• Same as (3)

(3) And (4) are Vacuously True or True by
Default by virtue of the fact that the
hypothesis is False
5
Other Ways to Say ?

• If p, then q
• p implies q
• if p, q
• p only if q
• p is sufficient for q
• q if p
• q whenever p
• q is necessary for p

• Not the same as the if-then construct used in
  programming languages

6
Related Implications

• Converse of
• p ? q
• is
• q ? p

p q p ? q q ? p
T T T T T F
F T F T T
F F F T T
Inverse of p ? q is p ? q
p q p ? q p?q
T T T T T F
F T F T T
F F F T T
Not Logically Equivalent!
7
Related Implications (contd)
Contrapositive of p ? q is the proposition
q ? p
p q p ? q q?p
T T T T T F
F F F T T
T F F T T
Logically Equivalent!
8
Practice
p You learn the simple things well q The
difficult things become easy

• You do not learn the simple things well.
• If you learn the simple things well then the
  difficult things become easy.
• If you do not learn the simple things well, then
  the difficult things will not become easy.

• The difficult things become easy but you did not
  learn the simple things well.
• You learn the simple things well but the
  difficult things did not become easy.

9
Practice
p You learn the simple things well q The
difficult things become easy

• You do not learn the simple things well. p
• If you learn the simple things well then the
  difficult things become easy. p ? q
• If you do not learn the simple things well, then
  the difficult things will not become easy.
  p ? q

• The difficult things become easy but you did not
  learn the simple things well. q
  ? p
• You learn the simple things well but the
  difficult things did not become easy.
  p ? q

10
Logical Equivalence with ?

• If you know that a statement p is true or that a
  statement r is true, you can deduce the truth of
  statement r by showing two things
• The truth of r follows from the truth of p and
• The truth of r follows from the truth of q
• Prove If Roland eats sweets or he eats potato
  chips then he will get fat
• p if Roland eats sweets
• q if Roland eats potato chips
• r he will get fat
• (p ? q) ? r ? (p ? r) ? (q ? r)

11
Rewriting If..Then as an Or

• If Roland eats sweets then he will get fat
• p ? q
• Roland does not eat sweets or he will get fat
• p ? q ? p ? q
• Also known as the Law of Implication
• If we negate ? we get
• (p ? q) ? p q
• Prove these with truth tables..

12
Biconditional

• Let p and q be propositions. The proposition p
  if and only if q is denoted p ? q
• True when p and q have the same truth values
• False otherwise
• p if and only if q means the same as.
• p only if q (q ? p) and p if q (p ? q)
• p is necessary and sufficient for q

13
Prove p ? q ? (p ? q) ? (q ? p)

• p q p ? q p ? q q ? p (p ? q) ? (q ?
  p)
• T T T T T T
• T F F F T F
• F T F T F F
• F F T T T T

We will call this the biconditional equivalence
14
Necessary and Sufficient

• Sufficient Condition
• The occurrence of r is sufficient to guarantee
  the occurrence of s
• r is a sufficient condition for s
• if r then s (r ? s)
• Necessary Condition
• The occurrence of r is necessary to obtain the
  occurrence of s
• r is a necessary condition for s
• if not r then not s (r ? s)

15
Necessary and Sufficient

• So to say r is a necessary and sufficient
  condition for s means
• (r ? s) ? (r ? s)
• (s ? r) ? (r ? s) contrapositive
• s ? r logical equivalence
• ? biconditional

16
Example

• p If it rains tomorrow
• q I will not go to town
• If (p ? q)
• If it rains tomorrow, then I will not go to town
• p is a sufficient condition for q
• Only if (q ? p)
• I will not go to town tomorrow only if it does
  rain
• p is a necessary condition for q